The emergence of fractal features in the microscopic structure of space-time is a common theme in many approaches to quantum gravity. In particular the spectral dimension, which measures the return probability of a fictitious diffusion process on space-time, provides a valuable probe which is easily accessible both in the continuum functional renormalization group and discrete Monte Carlo simulations of the gravitational action.

We relate the discrete classical phase space of loop gravity to the continuous phase space of general relativity. Our construction shows that the flux variables do not label a unique geometry, but rather a class of gauge-equivalent geometries. We resolve the tension between the loop gravity geometrical interpretation in terms of singular geometry, and the spin foam interpretation in terms of piecewise-flat geometry, showing that both geometries belong to the same equivalence class. We also establish a clear relationship between Regge geometries and the piecewise-flat spin foam geometries.

Tensor models appear as the higher dimensional extension of the so-called matrix models describing 2D quantum gravity through the sum over triangulations of surfaces. In the light of the recent $1/N$ expansion for these tensor models, we uncover a new class of tensor models for 4D and 3D gravity which are renormalizable at all orders of perturbation theory. An overview of two papers, [arXiv:1111.4997 [hep-th]] and [arXiv:1201.0176 [hep-th]], on the renormalization of these tensor models and their beta function will be given.

Group field theories show up as a higher dimensional generalization of matrix models in background independent approaches to quantum gravity.Their Feynman expansion generates simplicial complexes of all topologies weighted by spin foam amplitudes.In this talk, we will present a dual formulation of these theories as non-commutative quantum fields theories, whose variables have a clear interpretation in terms of simplicial geometry. We will show that it gives a geometrically clear ways to define spin foam models for gravity which can be cast as

To study the continuum limit of a microscopic model of gravity we need microscopic observables that have a clear interpretation in terms of continuum geometry. In general the construction of such observables is notoriously difficult. In the model of causal dynamical triangulations (CDT) it is clear what the microscopic observables are, but at present the only known well-behaved observables with a continuum interpretation are spatial volumes.

Entanglement is a paradigmatic example of quantum correlations, a presumed reason for the superior performance of quantum computation and an obvious divider of states and processes into classical and quantum. In the last decade all these notions were challenged. Entanglement does not capture the totality of non-classical behavior. Quantum discord (in its different versions) is a more general measure of quantum correlations. It can be related to the advantage in some tasks like the extraction of work from a Szilrad heat engine using Maxwell's demons with various resources.

We construct the q-deformed spinfoam vertex amplitude using Chern-Simons theory on the boundary 3-sphere of the 4-simplex. The rigorous definition involves the construction of Vassiliev-Kontsevich invariant for trivalent knot graph. Under the semiclassical asymptotics, the q-deformed spinfoam amplitude reproduce Regge gravity with cosmological constant at nondegenerate critical configurations.

In general relativity, the fields on a black hole horizon are obtained from those in the bulk by pullback and restriction. In quantum gravity, it would be natural to obtain them in the same manner. This is not fully realized in the quantum theory of isolated horizons in loop quantum gravity, in which a Chern-Simons phase space on the horizon is quantized separately from the bulk. I will outline an approach in which the quantum horizon degrees of freedom are simply components of the quantized bulk degrees of freedom.